Rockhopper is good for touring, semi slick tyres are ideal. Add a decent rear rack, bar ends and some mudguards for wet days. The default replacement saddle is Charge Spoon, most people like it; I use a very old, beat up Specialized one.
Mid Sept-Oct is a good time for France, probably dryer than March/April. S of France is OK for touring quite late into the season. Some campsites close for winter.
Consider travel to France by a budget airline to a regional airport in Brittany, it will be cheaper and quicker than rail/ferry/chunnel. Same for the return journey. Put all of your bags inside one package to minimize costs.
I have toured for a couple of weeks using a Gelhert Solo tent. Once you replace the steel pegs with aluminium, it is compact and lightweight and should last. It is quite small but at that price, you can't complain. Combine with a 3/4 length inflatable mat and a 3 season bag and you should be set.
You will need to cook on something, gas canister is easiest, alcohol the cheapest.
You could manage with 2 large panniers and a barbag. Altura is good; look for locking quick-release mounts, heel cutout profile, a flap or roll top (not zippers) and 1 external pocket. You may have to pile stuff on the rear rack. I use nylon drybags on the rack and inside one panniers.
The brand-name camping kit is expensive compared to discount stores like Go Outdoors or those mil-surplus places. Aldi and Lidl sometimes do good cycling gear at stupid prices. You will need shorts, helmet and gloves and lights. Cycling jersey is nice to have but you can use wicking T shirts. In France , Decathlon has good value cycling gear.
You can recharge at campsites. I have dynamo lighting but since LEDs came along, battery lights are good for touring.